Problem: Six 6-sided dice are rolled.  What is the probability that the number of dice showing even numbers and the number of dice showing odd numbers is equal?
Explanation: There are $\binom{6}{3}$ ways for 3 of the dice to show even numbers and 3 of them to show odd numbers.  Each roll is even with probability $\frac12$ and odd with probability $\frac12$, so each arrangement of 3 odd numbers and 3 even number occurs with probability $\left(\dfrac{1}{2}\right)^{\!6}$.  Thus, the probability that 3 dice out of 6 show even numbers is \[\binom{6}{3}\frac{1}{2^6}=\boxed{\frac{5}{16}}.\]